Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.Comment: 28 pages, sequel to Capturing Goodwillie's Derivative, arXiv:1406.042

Capturing Goodwillie's Derivative

Recent work of Biedermann and R\"ondigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the symmetric group on n objects. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.Comment: Final version, to appear. Substantially shortened from earlier version, with a significantly expanded introduction, new results and examples. 27 page

Unbased calculus for functors to chain complexes

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring. Much of the construction of the Taylor tower for functors to spectra carries over to this context. However, one of the essential steps in the construction requires proving that a particular functor is part of a cotriple. For this, one needs to prove that certain identities involving homotopy limits hold up to isomorphism, rather than just up to weak equivalence. As the target category of chain complexes is not a simplicial model category, the arguments for functors to spectra need to be adjusted for chain complexes. In this paper, we take advantage of the fact that we can construct an explicit model for iterated fibers, and prove that the functor is a cotriple directly. We use related ideas to provide concrete infinite deloopings of the first terms in the resulting Taylor towers when evaluated at the initial object in the source category.Comment: 20 page

Cosimplicial invariants and Calculus of homotopy functors

The origin of these investigations was the successful attempt by myself and coauthors to generalize rational equivalences of two constructions which suggest possible definitions of deRham cohomology of ???brave new??? rings, one of Rezk (using a cosimplicial resolution) and the other by Waldhausen (using a variant of Goodwillie???s Taylor tower). The proof of agreement of these constructions relies heavily on the fact that the functors involved take values in Spectra. Goodwillie conjectured the extension of these results to include the case of functors taking value in Spaces. The main result of my thesis is a proof of this conjecture (Theorem 7.1.2), using significantly different methods than in the stable setting of the joint work. This makes strong use of the intermediate constructions T_n F in Goodwillie???s Calculus of homotopy functors. I give a new model which naturally gives rise to a new family of towers filtering the Taylor Tower of a functor. I also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor F under general assumptions

Monadicity of the Bousfield–Kuhn functor

Let M n f be the localization of the ∞-category of spaces at the v n -periodic equivalences, the case n = 0 being rational homotopy theory. We prove that M n f is for n ≥ 1 equivalent to algebras over a certain monad on the ∞-category of T (n)-local spectra. This monad is built from the Bousfield– Kuhn functor